Prediction Model for the Environmental Noise Distribution of High-Speed Maglev Trains Using a Segmented Line Source Approach

Featured Application

The proposed noise prediction model provides a critical tool that urban planners can use to assess quantitatively the environmental impacts of elevated rail transit systems, including the effects of Maglev train noise on residential communities and potential disturbances to avian ecosystems. Thus, this model provides a scientific foundation for ecological conservation. The established modeling framework demonstrates significant applicability across multiple domains, including environmental protection, urban planning optimization, and intelligent transportation management systems.

Abstract

Based on the theory of uniform finite-length incoherent line source radiation and real vehicle online test data of Shanghai Maglev trains, a prediction model for environmental noise is established using an equivalent segmented line sound source approach. The noise produced by Shanghai high-speed Maglev trains running at speeds of 235, 300, and 430 km/h is tested and analyzed using microphones. The test data are combined with computational fluid dynamics simulations to divide the train’s sound sources equally into five sections. Theoretical calculations are carried out on the noise test data collected as the train passes by, and the source strength of each individual sub-sound source during the train operation is determined using the least-squares method. As a result, a prediction model for the environmental noise of high-speed Maglev trains, represented as a combination of multiple sources, is developed. The predicted results are compared with the measured values to validate the accuracy of the model. The proposed model can be used for environmental assessments before new train lines are launched, allowing for appropriate mitigation measures to be taken in advance to reduce the impact of Maglev noise on the surrounding residential and ecological environments.

1. Introduction

Global socioeconomic advancement has made rail transit systems an infrastructure priority, with the accelerated deployment of rail transit networks and emerging Maglev technologies, in particular. Although high-speed rail systems and Maglev technologies offer unparalleled velocity advantages, their environmental noise emission during operation has become a critical concern in international environmental protection protocols. This dual focus on transportation innovation and acoustic pollution mitigation has made Maglev-induced noise characterization a sustained research priority within the environmental engineering community.

Countries have established corresponding railway environmental noise prediction models based on their vehicles and line conditions. Germany proposed a corresponding prediction method for Schall 03 railway noise. This method discretizes the railway line into several small segments; each segment is simplified as a point sound source, and the noise contribution of each point sound source to the reception point is calculated. Finally, the total sound level is obtained using energy superposition. The German prediction model has a wide range of applications, but it involves considerably simplified processing and does not consider the time-domain characteristics of the sound source [1,2,3,4]. In the railway external noise prediction model proposed in Russia, a high-speed train is regarded as a finite long-line sound source, and the equivalent continuous sound pressure level (SPL) at the reference position is calculated using an empirical formula derived from the linear fitting of many test results [5]. The Russian prediction model has poor adaptability to different types of high-speed trains; the prediction method is relatively simple, and the time-domain characteristics of the sound source are not considered [6]. In the construction of Hokuriku Shinkansen in Japan, the high-speed railway noise prediction method (Hokuriku model) was adopted. It divides the noise sources into four categories from bottom to top: structure noise, wheel and rail noise under the vehicle, aerodynamic noise above the vehicle, and collector system noise. In the Hokuriku model, sound sources are classified according to the physical sound mechanism; however, the directivity of sound sources is not considered, and a method for determining the sound power level of each sub-sound source is not provided [7,8,9]. The United States of America proposed a railway noise prediction model for high-speed steel-wheel electric locomotives and electric multiple-unit (EMU) trains. In this method, the noise sources of train operation are divided into three categories: traction noise, wheel–rail noise, and aerodynamic noise. The sound exposure level (SEL) of each sound source radiating to the sound point is calculated. The American model is easy to use and has a wide range of applications; however, it does not consider the air absorption and time-domain characteristics of the sound source [10]. The railway environmental noise prediction method adopted in China is based on the “Technical guidelines for environmental impact assessment-Urban rail transit” issued by the Ministry of Ecology and Environment, which proposes two prediction models based on different train running speeds [11]. For railways and urban rail transit with speeds lower than 200 km/h, the noise source is regarded as an incoherent finite-length dipole line sound source, and the calculation is divided into two parts: the reference position noise radiation SPL and the noise correction term. For a railway with a speed of 200–350 km/h, the noise sources are divided into three types from bottom to top: lower wheel and rail noise, upper aerodynamic noise, and collector system noise. In this model, the sound power level is considered to be the acoustic parameter, the SPL radiated by each sub-sound source to the reception point is calculated, and the total SPL is obtained via energy superposition [12]. The sound power level of each sub-sound source is given by an empirical formula based on the differences in train operating speed and track type. The Chinese model proposed two prediction models based on different acoustic parameters according to the train running speed but did not provide a method of determining the reference source intensity at speeds below 200 km/h or above 350 km/h.

Research on the environmental noise prediction of high-speed railways is of great significance for proposing and improving environmental noise prediction models for high-speed Maglev trains. In 2012, the Federal Railroad Administration of the United States Department of Transportation revised the environmental noise prediction model for high-speed Maglev trains, which was similar to the railway noise prediction model. This prediction method is relatively mature and has high operability [10]. The research object of this prediction model is a TR08 Maglev train, which has a maximum operating speed of 480 km/h. The basic physical quantity of noise evaluation adopted by the American model is the SEL [13], and the reference position is 15.2 m from the centerline of the track. The environmental noise prediction model of high-speed Maglev trains proposed by the United States of America does not require any real vehicle test data, and the calculation process and physical meaning are clear. However, some limitations exist: this model does not consider the air absorption attenuation, and the sound barrier attenuation does not consider the noise frequency characteristics.

As the number of commercial high-speed Maglev trains worldwide is very limited, few studies have been conducted on their environmental noise prediction models. Therefore, in this study, the existing environmental noise prediction models [14,15] of high-speed Maglev trains and high-speed wheel–rail trains in various countries are combined using sound radiation theory, numerical simulation of the flow field, and real train online test data. An environmental noise prediction model is established; it can quickly predict the equivalent continuous SPL at different noise monitoring points outside the train when a Maglev train passes, predict and evaluate the impact of a high-speed Maglev train on the environment through noise before the construction of the line, enable these impacts to be treated and prevented in advance, and enable noise pollution to be avoided after the completion of the track line, affecting the ecology and living conditions in the area.

Considering the model development and validation procedure that constitute the central focus of this study, the remainder of the paper is organized as follows: Section 2 describes the noise source distribution of high-speed Maglev trains, details the noise time-domain characteristics and noise source distribution, and determines the five-part line source. In Section 3, the model and parameters of the noise source are constructed, and the intensity of the noise source at other speeds is fitted using the measured data. In Section 4, the validity of the model is verified by comparing the measured and predicted time-domain curves and equivalent continuous A-weighted sound level. Finally, Section 5 provides the conclusions of this study.

2. Analysis of High-Speed Maglev Noise Source Distribution

2.1. Numerical Simulation Model of the Flow Field

The three-dimensional and triple-unit configuration of the Shanghai TR08 Maglev system exhibits longitudinal dimensions of 27 m for both terminal vehicular modules and 24.5 m for the intermediate coach section, with 0.25 m windshield clearance between adjacent units, culminating in an aggregate length of 79 m for the complete train formation, as shown in Figure 1a. Geometric abstraction was used to solve the inherent surface irregularity problem in the geometry, and smaller auxiliary features were excluded from the computational domain. Vertical clearance and lateral span measurements were 4.7 and 3.7 m, respectively, while the transverse deck dimension of the guideway was 2.8 m, as shown in Figure 1b. The electromagnetic suspension maintained a 10 mm operational clearance above the elevated bridge structure, with a 10 m vertical elevation [16]. This streamlined topology modeled using SolidWorks is shown in Figure 1.

The fluid dynamics simulation domain encompassing the full-scale Maglev vehicle configuration on the guideway system is shown in Figure 2. The spatial arrangement comprised a 40 m inflow boundary offset from the leading vehicle nose and an 80 m outflow boundary clearance from the trailing vehicle nose, yielding a 199 m longitudinal computational span. Vertically, the domain extended 10 m below and 13.5 m above the guideway reference plane, establishing a total vertical dimension of 23.5 m. The lateral confinement of the domain measured 22.2 m in the transverse direction.

The aerodynamic simulation framework incorporated compressible flow considerations for Maglev velocities of 235, 300, and 430 km/h. The abcd boundary was configured as a pressure far-field with 1 atm pressure and a zero yaw angle. Symmetric constraints were applied to the aefb, bfgc, and cghd planes. Ground effect modeling utilized slip walls on aehd and guideway surfaces with velocities matching the inflows. The train surface was defined as a no-slip stationary wall. The efgh boundary implemented a pressure outlet at ambient conditions. Two mesh refinement zones (Blocks 1 and 2) were strategically positioned around the train geometry to ensure computational accuracy, as shown in Figure 2.

The computational framework employed a hybrid meshing strategy with refinement zones to achieve the required solution accuracy. The mesh partitioned using Fluent Meshing is shown in Figure 3. Surface discretization followed curvature-adaptive principles where grid element dimensions scaled inversely with surface curvature gradients. Critical flow resolution parameters were defined through simultaneous resolution of boundary layer dynamics and turbulence modeling requirements. Specific element sizing included 10 mm for aerodynamically sensitive zones (streamlined nose/tail sections), 30 mm for geometrically stable regions (door interfaces, carriage junctions, and underbody components), and 100 mm for guideway surfaces. The final surface grid configuration contained approximately 22 million elements.

The near-body flow region was discretized using unstructured prismatic elements to accommodate complex geometries. A wall-resolved large eddy simulation (LES) approach was implemented with the first grid node placement at 0.01 mm normal to the solid surfaces. The boundary layer transition was achieved through 25 progressively expanding prismatic layers (growth factor 1.05), ensuring a smooth velocity gradient resolution between the viscous sublayer and main flow regions.

The meshing strategy employed dual refinement blocks to enhance the flow-resolution accuracy. Block 1 enveloped the Maglev train surfaces with a maximum element size of 100 mm, while Block 2 had a 300 mm mesh control to mitigate wake interference. Transitional mesh layers bridged prismatic boundary elements with hexahedral core grids, maintaining global element dimensions <500 mm. The final volumetric grid configuration contained approximately 255 million elements, with the mesh topology of the symmetry plane shown in Figure 4.

A dual-phase computational strategy implemented in ANSYS Fluent ensured numerical stability in the transient flow analysis. Initial steady-state simulations (3000 iterations) employed compressible shear stress transport (SST) k-ω turbulence modeling [17], solved via a semi-implicit pressure-linked equations (SIMPLE) algorithm with second-order upwind discretization for momentum/turbulence parameters. Converged steady-state solutions provided initial conditions for transient LES computations utilizing the wall-adapting local eddy (WALE) sub-grid model. Temporal discretization adopted a bounded second-order implicit schemes, while spatial terms used central differencing for momentum with standard pressure-velocity coupling.

A two-stage temporal resolution strategy was implemented for the flow field evolution analysis. The initial phase employed a timestep of 1 × 10⁻⁴ s to advance the solution to 0.1 s, ensuring turbulent flow development. Subsequently, the temporal resolution was enhanced to 5 × 10⁻⁵ s with full-field data archiving per timestep, extending the simulation to 0.2 s for capturing detailed vortex dynamics.

2.2. On-Track Experiment

Measurements were obtained under naturally occurring conditions characterized by ambient wind speeds <1.5 m/s and background noise levels <60 dB. Operational velocities spanning 230–430 km/h at a zero yaw angle were systematically evaluated. Figure 5 shows the acoustic measurement array configuration for pass-by noise characterization.

The height of measuring points X1–X6 above the ground was 1.2 m, and the horizontal distances of X1–X6 from the center line of the guideway were 7.5, 10, 15, 25, 45, and 90 m, respectively. The horizontal distance of measuring point Y1 from the centerline of the guideway was 25 m, and the height of the guideway surface was 3.5 m. Referring to the international standard ISO3095-2013, this measuring point is called the noise standard point [18]. All microphones were in the same cross-section perpendicular to the guideway, and the on-site pass-by noise testing setup is shown in Figure 6.

The experimental setup incorporated Brüel & Kjær LAN-XI hardware with PULSE LabShop software, operating at a 65.5 kHz sampling rate across a 0–25.6 kHz frequency range. This configuration supported extended-duration signal acquisition while performing fast Fourier transform (FFT) and constant percentage bandwidth (CPB) analyses.

2.3. Analysis of Simulation and Experimental Results

Figure 7 compares the time-domain curves of the pass-by noise at measuring point Y1 at different train operating speeds, the vertical axis denotes the A-weighted fast time weighted sound pressure level L_A,F. During the passage of the Maglev train, the L_A,F initially exhibited a rapid increase and maintained relative stability over a brief duration, followed by a gradual decrease, with the curve demonstrating asymmetric characteristics.

Figure 8 shows the equivalent continuous A-weighted SPL L_Aeq,T variation curves in logarithmic coordinates with lateral distance at measurement points X1–X6 at different train operating speeds. The L_Aeq,T continuously decreased with increasing lateral distance, with essentially consistent attenuation patterns across all tested speeds. A strong linear relationship was observed between the L_Aeq,T and logarithm of the lateral distance. The fitted curves for each train operating speed exhibited coefficients of determination >0.99, with attenuation coefficients ranging from 12.3 to 14.8 dB/log₁₀(m). The measured sound level reduction per distance doubling reached 3.7–4.5 dB(A), indicating that the external noise radiation characteristics of the Maglev train correspond to those of an incoherent line source.

Figure 9 shows the numerically simulated three-dimensional vortex structures visualized using the velocity-colored Q-criterion around the Maglev train operating at 430 km/h. The results reveal that (1) vortex formation occurred at the high-pressure zone of the leading train owing to premature airflow separation; (2) the streamlined profiles induced flow separation at both the head and tail sections, generating distinct detached vortex structures downstream; and (3) a pair of counter-rotating wake vortices developed at the tail section, evolving into stable trailing vortex structures during downstream propagation. With increasing train operating speed, the vortex distribution demonstrated enhanced spatial complexity, exhibiting larger spanwise/vertical scales of vortex pairs, which expanded the effective noise source distribution range. Therefore, regions with intense turbulent movement were taken as strong noise sources. As shown in Figure 9, the positive-pressure area in front of the train, the streamline part of the head, the middle part of the train, the streamline part of the tail, and the negative-pressure area in the rear of the train were divided into five parts: l₁, l₂, l₃, l₄, and l₅.

3. Construction of the Noise Prediction Model

3.1. Finite-Length Incoherent Moving Line Source Model

Figure 10 provides a schematic of the acoustic radiation from a finite-length moving line source within the global coordinate system O-xyz. The line source moves along the positive x-axis at velocity v₀, with the reception point R located at global coordinates (0, d₀, h₀) and maintaining a perpendicular distance r₀ from the source axis. The blue cylinder represents the infinitesimal element of a line source; its coordinates are (x′, 0, 0), and the radiated acoustic power is w(x′)dx′. τ is the moment at which the line source element emits sound waves, t is the moment at which R receives the sound waves emitted by the line source element at time τ, and the center of the line source moves exactly to point O at time t = 0. The spatial global x-axis coordinates of the element at times τ and t are x′ + v₀τ and x′ + v₀t, respectively. The distance between the element and R is r, and the angles between the lines from the element to R and the x-, y-, and z-axes are φ, θ, and γ, respectively.

r is the distance that the sound wave travels at the sound speed c₀, which can be expressed as

$$r=c_0(t-\tau )=\sqrt{{(x^{\prime }+v_0\tau )}^2+r_0^2}$$

(1)

After squaring Equation (1), it can be arranged into a quadratic equation about τ:

$$(1-M_0^2){\tau }^2-2\left(t+{\displaystyle \frac{M_0{x}^{\prime }}{c_0}}\right)\tau +t^2-{\displaystyle \frac{{x^{\prime }}^2+r_0^2}{c_0^2}}=0,$$

(2)

where the Mach number M₀ = v₀/c₀ < 1, and the following solution for τ can be obtained:

$$\tau ={\displaystyle \frac{M_0{x}^{\prime }+c_{0}t\pm \sqrt{{(x^{\prime }+v_{0}t)}^2+(1-M_0^2)\cdot r_0^2}}{c_0(1-M_0^2)}}.$$

(3)

Substituting the above formula into Equation (1) obtains

$$r=c_0(t-\tau )={\displaystyle \frac{-M_0(x^{\prime }+v_{0}t)\mp \sqrt{{(x^{\prime }+v_{0}t)}^2+(1-M_0^2)\cdot r_0^2}}{1-M_0^2}}.$$

(4)

As $\sqrt{{(x^{\prime }+v_{0}t)}^2+(1-M_0^2)\cdot r_0^2}>\left|x^{\prime }+v_{0}t\right|>\left|-M_0(x^{\prime }+v_{0}t)\right|$, r is always greater than 0 when a positive sign is taken before the root of the upper formula, and r is always less than 0 when a negative sign is taken. As r is a distance, it only has meaning when r is a positive real number and a positive sign is taken before the root of the upper formula. If the directivity of the sound source is not considered, the square of the sound pressure generated by the sound wave emitted by the microelement propagating to R in the free field is [19,20]

$$\mathrm{d}{p^{\prime }}^2={\displaystyle \frac{\rho c_{0}w(x^{\prime })}{4\pi r{(x^{\prime })}^2}}\mathrm{d}{x}^{\prime }={\displaystyle \frac{\rho c_{0}w(x^{\prime }){(1-M_0^2)}^2}{4\pi {\left(\sqrt{{(x^{\prime }+v_{0}t)}^2+(1-M_0^2)\cdot r_0^2}-M_0\cdot (x^{\prime }+v_{0}t)\right)}^2}}\mathrm{d}{x}^{\prime }.$$

(5)

When a Maglev train runs at high speed, the fluid on the boundary is subjected to pulsating pressure on the surface of the car body, and the pulse power source can be regarded as a dipole sound source whose dipole moment direction is consistent with the side normal direction of the car body. If a line source element is considered to have dipole directivity, then the square of the sound pressure propagated by the sound wave emitted by the element to reception point R can be expressed as

$$\mathrm{d}{p^{\prime }}^2={\displaystyle \frac{\rho c_{0}w(x^{\prime }){\cos }^2\theta }{4\pi r{(x^{\prime })}^2}}\mathrm{d}{x}^{\prime }={\displaystyle \frac{\rho c_0{d}_0^{2}w(x^{\prime })}{4\pi r{(x^{\prime })}^4}}\mathrm{d}{x}^{\prime },$$

(6)

where $\cos \theta =d_0/r$. Without the loss of generality, the directivity factor of $\mathrm{d}{p^{\prime }}^2$ can be expressed as ${\cos }^{2m}\theta$, and m values of 0, 0.5, and 1, respectively, yield the directivity of no direction, intermediate directivity, and dipole directivity. Then, the square of the sound pressure of the sound wave propagated by the element to R is

$$\mathrm{d}{p^{\prime }}^2={\displaystyle \frac{\rho c_{0}w(x^{\prime }){\cos }^{2m}\theta }{4\pi r{(x^{\prime })}^2}}\mathrm{d}{x}^{\prime }={\displaystyle \frac{\rho c_0{d}_0^{2m}w(x^{\prime })}{4\pi r{(x^{\prime })}^{2+2m}}}\mathrm{d}{x}^{\prime }.$$

(7)

If the sound source is a uniform and incoherent line source, then the square of the sound pressure at R at time t is the linear superposition [21] of the squares of the sound pressures when the sound waves emitted by each element reach the receiving point at time t, so the SPL at R at time t can be expressed as

$$\begin{array}{l}SPL(t)=10{\log }_{10}\left(p^{\prime }{(t)}^2/{p^{\prime }}_{ref}^2\right)\\ =10{\log }_{10}\left({\displaystyle \underset{-l/2}{\overset{l/2}{\int }}\frac{\rho c_{0}w{d}_0^{2m}{(1-M_0^2)}^{2+2m}}{4\pi {p^{\prime }}_{ref}^2{\left(\sqrt{{(x^{\prime }+v_{0}t)}^2+(1-M_0^2)\cdot r_0^2}-M_0\cdot (x^{\prime }+v_{0}t)\right)}^{2+2m}}}\mathrm{d}{x}^{\prime }\right)\end{array},$$

(8)

where ${p^{\prime }}_{ref}$ is the reference sound pressure, with a value of 20 μPa, and l is the length of the line source. A moving object causes a stress change in the surrounding fluid, which can be regarded as a pair of equal and opposite dipoles acting on the same spatial position, that is, a quadrupole sound source. If the linear sound source element is considered to have equiaxial quadrupole directivity and the plane formed by a pair of dipoles that make up the quadrupole is parallel to the y–z plane, then the square of the sound pressure propagated by the sound wave emitted by the element to the reception point can be expressed as

$$\mathrm{d}{p^{\prime }}^2={\displaystyle \frac{\rho c_{0}w(x^{\prime }){\cos }^2\theta {\cos }^2\gamma }{4\pi r{(x^{\prime })}^2}}\mathrm{d}{x}^{\prime }={\displaystyle \frac{\rho c_0{h}_0^2{d}_0^{2}w(x^{\prime })}{4\pi r{(x^{\prime })}^6}}\mathrm{d}{x}^{\prime }.$$

(9)

Similarly, the SPL at the reception point at time t can be expressed as

$$\begin{array}{ll}SPL(t)& =10{\log }_{10}\left(p^{\prime }{(t)}^2/{p^{\prime }}_{ref}^2\right)\\ & =10{\log }_{10}\left({\displaystyle \underset{-l/2}{\overset{l/2}{\int }}\frac{\rho c_{0}w{h}_0^2{d}_0^2{(1-M_0^2)}^6}{4\pi {p^{\prime }}_{ref}^2{\left(\sqrt{{(x^{\prime }+v_{0}t)}^2+(1-M_0^2)\cdot r_0^2}-M_0\cdot (x^{\prime }+v_{0}t)\right)}^6}}\mathrm{d}{x}^{\prime }\right)\end{array}$$

(10)

3.2. Equivalent Source Strength

Based on the simulation results of the flow field described in Section 2.3, the exterior noise prediction of the high-speed Maglev train was implemented using a five-segment incoherent line source equivalent model. The sound sources were equivalently modeled as finite-length incoherent line sources with intermediate directivity, where the sound power levels L_w1, L_w2, L_w3, L_w4, and L_w5 correspond to the front positive-pressure zone, streamlined head section, middle carriage section, streamlined tail section, and rear negative-pressure zone, respectively. Computational validation confirmed the optimal positioning of line source segments at the rail level (0 m height) on the vehicle side, with the horizontal distance between R and the vehicle side defined as d. Figure 11 provides a schematic of the equivalent acoustic source model for the high-speed Maglev train noise prediction.

Reference standards ISO 3095-2013 [18] and HJ 2.4-2021 [11] define the position 25 m from the centerline of the guideway and 3.5 m above the guideway surface as the reference measurement point of the source intensity. The equivalent continuous A-weighted sound level during the train passing time at this position is called the noise source intensity. When sound sources S₁–S₅ act alone, the noise source intensity is L_p1–L_p5. As the noise source intensity obtained in a real train test is the result of the joint action of all sound sources, to improve the accuracy of the environmental noise prediction model of the Maglev train, L_p1–L_p5 should be separated.

According to Equation (8), the SPL at the reception point generated by the combined effect of five uniform finite-length line source segments with intermediate directivity is expressed as

$$\begin{array}{l}{L}_p(t)=10{\log }_{10}\left({\displaystyle \frac{d{(1-M_0^2)}^3}{4\pi }}\right)+\\ 10{\log }_{10}\left(\begin{array}{c}{\displaystyle \underset{v_{0}t-\frac{L}{2}}{\overset{v_{0}t-\frac{L}{2}+l_1}{\int }}\frac{{10}^{0.1L_{w1}}}{{\left(\sqrt{x^2+(1-M_0^2)\cdot r_1^2}-M_{0}x\right)}^3}\mathrm{d}x+}{\displaystyle \underset{v_{0}t-\frac{L}{2}+l_1}{\overset{v_{0}t-\frac{l_3}{2}}{\int }}\frac{{10}^{0.1L_{w2}}}{{\left(\sqrt{x^2+(1-M_0^2)\cdot r_2^2}-M_{0}x\right)}^3}\mathrm{d}x+}\\ {\displaystyle \underset{v_{0}t-\frac{l_3}{2}}{\overset{v_{0}t+\frac{l_3}{2}}{\int }}\frac{{10}^{0.1L_{w3}}}{{\left(\sqrt{x^2+(1-M_0^2)\cdot r_3^2}-M_{0}x\right)}^3}\mathrm{d}x+}{\displaystyle \underset{v_{0}t+\frac{l_3}{2}}{\overset{v_{0}t+\frac{l_3}{2}+l_4}{\int }}\frac{{10}^{0.1L_{w4}}}{{\left(\sqrt{x^2+(1-M_0^2)\cdot r_4^2}-M_{0}x\right)}^3}\mathrm{d}x+}\\ {\displaystyle \underset{v_{0}t+\frac{l_3}{2}+l_4}{\overset{v_{0}t+\frac{L}{2}}{\int }}\frac{{10}^{0.1L_{w5}}}{{\left(\sqrt{x^2+(1-M_0^2)\cdot r_5^2}-M_{0}x\right)}^3}\mathrm{d}x}\end{array}\right)\end{array}.$$

(11)

The total length of the five-segment line source is denoted as L, where l₁–l₅ represent the lengths of the line source segments, that is, $L=l_1+l_2+l_3+l_4+l_5$, where l₂ = l₄ = 7 m, l₃ = 65 m, and the sum of the three segments provides a total vehicle length of 79 m. As can be seen from Figure 12, there is a good linear relationship between the length of the line source and the speeds of l₁ and l₅, where $l_1=57.15-0.18\cdot v$, $l_5=35.00+0.38\cdot v$, and v is the train operating speed to be predicted in m/s.

Designating the reference measurement position as R, the instantaneous SPL L_p,R(t_i) of the Shanghai Maglev train pass-by noise at R were measured with 0.1 s time intervals based on the full-scale test data from Section 2.2. Using Equation (11), the predicted instantaneous SPL $\tilde{L}$_p,R(t_i) at R can be calculated when specific values are assigned to the sound power levels per unit length L_w1–L_w5. A least-squares optimization framework was implemented with L_w1–L_w5 as the optimization parameters, aiming to minimize the total mean-squared error between the measured and predicted instantaneous SPLs at R.

The optimization process was conducted with three distinct step sizes (10, 1, and 0.1 dB(A)) and corresponding variation ranges (±30, ±10, and ±5 dB(A)) for parameter adjustment. By systematically exploring all possible values within the 65–125 dB(A) range, the total mean squared error between the measured and predicted instantaneous SPLs at R was calculated for each combination of parameters. The L_w1–L_w5 values that achieved the optimization objective were identified as the optimal sound power levels per unit length. This procedure was repeated for Maglev train operating speeds of 235, 300, and 430 km/h to determine the optimal speed-dependent L_w1–L_w5 values, as shown in

Table 1. Optimal values of L_w1–L_w5 at different train operating speeds (dB(A)).

Train Operating Speed (km/h)	Lw1	Lw2	Lw3	Lw4	Lw5
235	76.3	108.2	110.1	111.5	100.5
300	83.6	110.8	114.7	115.3	107.3
430	94.6	115.4	120.9	120.8	118.2

.

The optimal values of L_w1–L_w5 at different train operating speeds can be used in Equation (11) to calculate the instantaneous SPL at R. As can be seen from

Table 2. Measured versus predicted SPL errors at different train operating speeds (dB(A)).

Train Operating Speed (km/h)	Measured LAeq,E	Predicted LAeq,P	LAeq Error |Δ|
235	87.6	87.9	0.2
300	91.8	92.2	0.4
430	98.1	98.6	0.5

, further calculation results show that the equivalent continuous A-weighted sound level L_Aeq was obtained during the train passing time, and the absolute error between the measured L_Aeq,E and predicted L_Aeq,P values was <0.4 dB(A). These error metrics confirm the high accuracy of the L_w1–L_w5 values derived through the least-squares optimization method, validating the effectiveness of the proposed line source model for Maglev train noise prediction across multiple operating speeds. After L_w1–L_w5 were solved, according to Equation (11), the noise source intensities L_p1–L_p5 when the sound sources S₁–S₅ acted alone were solved.

Figure 13 shows the variation in the sound power levels L_w1–L_w5 per unit length of the line sources with train operating speed, where the reference speed v_ref was 235 km/h. It can be seen from Figure 13 that the sound power levels per unit length of the line sources have a good linear relationship with the normalized velocity logarithm. The prediction formulae for L_w1–L_w5 can be expressed as

$$\left\{\begin{array}{c}{L}_{w1}=69.8{\log }_{10}\left({\displaystyle \raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$v_{ref}$}\right.}\right)+76.3\\ L_{w2}=27.6{\log }_{10}\left({\displaystyle \raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$v_{ref}$}\right.}\right)+108.1\\ L_{w3}=41.0{\log }_{10}\left({\displaystyle \raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$v_{ref}$}\right.}\right)+110.2\\ L_{w4}=35.4{\log }_{10}\left({\displaystyle \raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$v_{ref}$}\right.}\right)+111.5\\ L_{w5}=67.6{\log }_{10}\left({\displaystyle \raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$v_{ref}$}\right.}\right)+100.4\end{array}\right.,$$

(12)

3.3. Prediction Model and Parameters

Drawing on the theoretical framework of incoherent finite-line source radiation from Section 3.1, combined with Shanghai Maglev train field measurement data (Section 2.2) and aerodynamic simulation results (Section 2.3), established international methodologies for the exterior noise prediction of high-speed wheel–rail trains and Maglev systems were synthesized to develop an environmental noise prediction model for high-speed Maglev trains. The proposed model employs a five-segment line source equivalence approach, with the equivalent continuous A-weighted sound level L_Aeq,T at the reception points given by

$$L_{Aeq,T}=10{\log }_{10}\left[{\displaystyle \frac{1}{T}}\left({\displaystyle \sum _i{t}_{eq,i}\cdot {10}^{0.1L_{p,i}}}\right)+{10}^{0.1L_b}\right],$$

(13)

where T is the evaluation time, L_b is the equivalent continuous A-weighted sound level of the background noise at the prediction point, t_eq,i is the equivalent passing time of train i, and t_eq,i is calculated as follows:

$$t_{eq,i}={\displaystyle \frac{L}{v}}\left(\sqrt{1+r^2/L^2}+r/L\right),$$

(14)

where v is the train operating speed to be predicted, in m/s, and r is the distance from the predicted point to the line source. L_p,i is the equivalent continuous A-weighted sound level during passage of the i-th train. The formula is as follows:

$$L_{p,i}=10{\log }_{10}\left[{10}^{0.1\left(L_{p1,i}+C_i\right)}+{10}^{0.1\left(L_{p2,i}+C_i\right)}+{10}^{0.1\left(L_{p3,i}+C_i\right)}+{10}^{0.1\left(L_{p4,i}+C_i\right)}+{10}^{0.1\left(L_{p5,i}+C_i\right)}\right],$$

(15)

where L_p is the noise source strength of train i at the reference point, which can be obtained from Section 3.2.

$$C_i=C_{v,i}+C_{div,i}+C_{air,i}+C_{g,i},$$

(16)

where C_v,I, C_div,I, C_air,I, and C_g,i are the speed correction, geometric divergence correction, air absorption correction, and ground correction of the i-th train, respectively.

For speed correction, according to the fitting formula of the sound power level per unit length of the line source with the running speed, the calculation formula for C_v,i is as follows [11]:

$$C_{v,i}=10{\log }_{10}\left({\displaystyle \raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$v_{ref}$}\right.}\right),$$

(17)

where v_ref is the reference speed, which is 235 km/h.

For the geometric divergence correction, the geometric divergence correction of the i-th train can be expressed as

$$C_{div,i}=10 {\log }_{10}\left({\displaystyle \frac{d{r}_0^3\left(1+\sqrt{1+L^2/r_0^2}\right)}{d_0{r}^3\left(1+\sqrt{1+L^2/r^2}\right)}}\right),$$

(18)

where d₀ and d are the horizontal distances from the strong source reference point and predicted point, respectively, to the side of the car body, and r₀ and r are the distances from the strong source reference point and predicted point, respectively, to the sound source.

With air absorption attenuation, sound energy is generated during the heat energy dissipation process owing to the medium viscosity, heat conduction, and molecular relaxation absorption effect of sound waves in air. Sound absorption in air is related to the temperature, humidity, frequency, and propagation distance, and the atmospheric absorption correction of the i-th train can be uniformly expressed as

$$C_{air,i}=-\alpha (r-r_0).$$

(19)

In Equation (18), α is the attenuation coefficient of air absorption, and

Table 3. Octave-band air absorption attenuation coefficient (units: dB/m).

Temperature (°C)	Relative Humidity (%)	Octave-Band Center Frequency (Hz)
		125	250	500	1000	2000	4000
25	50	3.99 × 10−4	1.32 × 10−3	3.23 × 10−3	5.68 × 10−3	1.02 × 10−2	2.57 × 10−2
25	60	3.40 × 10−4	1.18 × 10−3	3.18 × 10−3	5.96 × 10−3	1.02 × 10−2	2.32 × 10−2
25	70	2.96 × 10−4	1.06 × 10−3	3.08 × 10−3	6.19 × 10−3	1.04 × 10−2	2.19 × 10−2
35	60	2.57 × 10−4	9.77 × 10−4	3.32 × 10−3	8.45 × 10−3	1.51 × 10−2	2.58 × 10−2
15	60	4.26 × 10−4	1.18 × 10−3	2.31 × 10−3	4.06 × 10−3	9.5 × 10−3	3.03 × 10−2
0	60	4.01 × 10−4	7.79 × 10−4	1.78 × 10−3	5.50 × 10−3	1.93 × 10−2	6.33 × 10−2
–10	60	3.60 × 10−4	9.69 × 10−4	3.23 × 10−3	1.09 × 10−2	2.96 × 10−2	5.35 × 10−2

presents the attenuation coefficients of air absorption in the octave band under partial temperature and humidity conditions [22]. In general, the sound attenuation caused by air absorption is relatively small, with high-frequency sound waves experiencing greater absorption attenuation than low-frequency sound waves.

The ground correction factors mainly include the correction C_g,r,i caused by ground reflection and ground attenuation correction C_g,a,i caused by interference between direct and reflected sound paths. For the i-th train, the ground correction is expressed as [11]

$$C_{g,i}=C_{g,r,i}+C_{g,a,i}.$$

(20)

The SPL at R is the result of the superposition of the direct sound and ground-reflected sound energy, and r_d and r_r are the distances from the sound source S to R and image point R′, respectively. The values of C_g,r,i for smooth, flat, and hard ground are shown in

Table 4. Ground reflection corrections [23] (units: dB(A)).

rr/rd	≈1	≈1.4	≈2	>2.5
Cg,r,i	3	2	1	0

.

For loose ground or mixed ground that is mostly loose, the ground attenuation correction C_g,a,i is calculated as follows [11]:

$$C_{g,a,i}=\left(2h_m/r\right)\cdot \left(17+300/r\right)-4.8$$

(21)

In Equation (20), h_m is the average ground height of the propagation path. If C_g,a,i calculated according to Equation (20) is positive, then zero is used to replace C_g,a,i [11].

4. Model Verification and Noise Prediction

To validate the accuracy of the proposed environmental noise prediction model for Maglev trains, a comparative analysis was conducted between the field measurements and model predictions at monitoring points 1.2 m above the ground level along the Shanghai Maglev line. Figure 14 presents time-domain comparisons of the measured and predicted A-weighted fast time weighted sound pressure level L_A,F at 45 m from the guideway centerline (1.2 m height) under different train operating speeds. Close alignment was observed between the predicted and measured time-domain trends at 235, 300, and 430 km/h.

The predicted time-domain noise curve of a Maglev train passing at 600 km/h was analyzed at a measurement point located 45 m from the guideway centerline and 1.2 m above ground level. As illustrated in Figure 15, while the temporal characteristics maintained similar trends across different speeds, the 600 km/h speed exhibited significantly sharper peak amplitudes in the waveform compared to those of lower speeds. Furthermore, Figure 15 presents the equivalent continuous A-weighted sound level (L_Aeq) contributions from individual source intensities (L_P1–L_P5) during train passage at 600 km/h.

Flow field analysis revealed that acoustic energy was primarily concentrated in two critical regions: (1) the near-surface boundary layer along the train body and (2) the wake flow zone downstream of the trailing vehicle. The aerodynamic noise generation mechanisms can be attributed to two dominant flow phenomena: (1) multiple separation vortices along the streamlined shoulder of the leading car, induced by flow separation at the critical geometric transition, and (2) large-scale trailing vortices extending over considerable distances behind the trailing car, resulting from similar flow separation mechanisms at its shoulder region. These observations confirm that vortex-dominated flow structures serve as the primary acoustic sources in high-speed Maglev operations. The intensified vortex dynamics at 600 km/h significantly amplified both peak noise levels compared to lower train operating speeds.

Figure 16 compares the measured and predicted equivalent continuous A-weighted SPL attenuation profiles in logarithmic coordinates during Maglev train pass-by events across multiple speeds as a function of the lateral distance from the guideway centerline (1.2 m height). The experimental and modeled decay trends at identical speeds agree closely, showing consistent distance-dependent attenuation. The cross-speed analysis results reveal maximum absolute deviations of 1.3 dB(A) across all ground-level monitoring points, validating the accuracy of the model while providing extrapolated profiles for 600 km/h operation.

5. Conclusions

In this study, a segmented line source model was developed for predicting the environmental noise distributions of high-speed Maglev trains by integrating uniform finite-length incoherent moving line source (UFL-IMLS) radiation theory with numerical simulations of the external flow field of a Shanghai Maglev train. The key findings are as follows:

(1)

Flow Field Analysis: Incorporating air compressibility effects, the LES method was employed to conduct numerical simulations of the external flow field surrounding a Shanghai Maglev train. The analysis revealed distinct three-dimensional vortical structure distributions with regions of elevated vorticity magnitude concentrated at the vehicle surface and wake zone. Flow separation was observed at the streamlined shoulder of the trailing car, whereas the wake region exhibited a complex array of multiscale vortices with varying intensities, dominated by a pair of counter-rotating ribbon-like trailing vortices. The vortex was the main sound source of fluid flow, and the noise energy was mainly concentrated in the turbulent movement area, such as near the surface of the vehicle body and the wake area. Therefore, the five parts of the turbulent movement around the train were taken as the five-segment line sources.

(2)

Model Validation: The results of a five-segment UFL-IMLS equivalent model derived from the flow simulation data demonstrated strong agreement with the on-site measurements. The maximum absolute deviations in the equivalent sound levels across the monitoring points were constrained to 1.3 dB(A), satisfying the requirements of engineering predictions.

(3)

High-Speed Extrapolation: The validated predictions were extended to 600 km/h operation, generating lateral attenuation curves of equivalent continuous A-weighted SPLs and time-domain profiles and providing critical reference information for ultra-high-speed Maglev noise assessment. The time-domain curves of each velocity had similar trends, and the time-domain curve peaks became sharper with an increase in velocity. The curve trends of equivalent continuous A-weighted SPLs with the lateral distance were the same at each speed, and at 600 km/h, the equivalent continuous A-weighted SPLs decreased by 11.8 dB as the logarithm of the lateral distance doubles.